Related papers: On the Dirichlet problem for special Lagrangian cu…

This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the…

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for incompressible MHD equations in a bounded smooth domain of three spatial dimensions with initial density being allowed to have…

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical…

The convexity of solutions to boundary value problems for fully nonlinear elliptic partial differential equations (such as real or complex $k$-Hessian equations) is a challenging topic. In this paper, we establish the power convexity of…

We study the Dirichlet problem for the weighted Schr\"odinger operator \[-\Delta u +Vu = \lambda \rho u,\] where $\rho$ is a positive weighting function and $V$ is a potential. Such equations appear naturally in conformal geometry and in…

We establish the Pogorelov type estimates for degenerate prescribed k-curvature equations as well as k-Hessian equations. Furthermore,we investigate the interior C1,1 regularity of the solutions for Dirichlet problems. These techniques also…

This article is concerned with the second boundary value problem of the Lagrangian mean curvature type equation arising from special Lagrangian geometry. By the parabolic method, we consider a fully nonlinear parabolic equation with oblique…

In this short paper we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below).This condition is related to a celebrated result of Milnor that classifies parabolic…

A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods…

Under structural conditions which are almost optimal, we derive a quantitative version of boundary estimate then prove existence of solutions to Dirichlet problem for a class of fully nonlinear elliptic equations on Hermitian manifolds.

In the paper obtained equivalent system of Fredholm integral equations in the study of the Dirichlet problem for the generalized Manjeron equation with non-smooth coefficients in non-classical treatment (1), (4). When non-smooth conditions…

In this work, we develop an efficient solver based on neural networks for second-order elliptic equations with variable coefficients and singular sources. This class of problems covers general point sources, line sources and the combination…

The work treats smoothing and dispersive properties of solutions to the Schrodinger equation with magnetic potential. Under suitable smallness assumption on the potential involving scale invariant norms we prove smoothing - Strichartz…

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…

The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping damping term of the form $(-\Delta)^\alpha\partial_t u$, $\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\Bbb R^3$. It appears that to…

This paper is devoted to derive some necessary and suficient conditions for the existence of positive solutions to a singular second order system of dynamic equations with Dirichlet boundary conditions. The results are obtained by employing…

We propose a monotone, and consistent numerical scheme for the approximation of the Dirichlet problem for the normalized Infinity Laplacian, which could be related to the family of so--called two--scale methods. We show that this method is…

We prove a result of existence of positive solutions of the Dirichlet problem for $-\Delta_p u=\mathrm{w}(x)f(u,\nabla u)$ in a bounded domain $\Omega\subset\mathbb{R}^N$, where $\Delta_p$ is the $p$-Laplacian and $\mathrm{w}$ is a weight…

In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori…

Given a C2-domain with compact boundary in an arbitrary complete Riemannian manifold, we search for smallness conditions on the boundary data for which the Dirichlet problem for the minimal hypersurface equation is solvable. We obtain an…